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NEW YORK UNIVERSITY COURANT INSTITUTE - LIBkARY
IMM-NYU 335 JANUARY, 1965
NEW YORK UNIVERSITY COURANT INSTITUTE OF MATHEMATICAL SCIENCES
FREDHOLM OPERATORS AND THE ESSENTIAL SPECTRUM
MARTIN SCHECHTER
PREPARED UNDER
CONTRACT NO. DA31-124-AR0- D-204
WITH THE
U. S. ARMY RESEARCH OFFICE
TI^™I-NYU 355
Courant Institute of Mathematical Sciences Nev; York Un5.verslty
PREDHOLM OPERATORS AND THE ESSENTIAL SPECTRUI4
Contract No. DA31-12^-AR0".D-204 January 1955
Martin Schechter
Department of Army Project No. 20011501B70^
"Request for additional copies by Agencies of the Department of Defense, their contractors, and other Government agencies should be directed to:
Defense Documentation Center Cameron Station Alexandria, Virginia 2231H
Department of Defense contractors must have established for DDC services or have their "need-to-know" certified by the cognizant military agencies of their project or contract."
Reproduction in I'jhole or in part is permitted for any purpose of the United States Government.
tii^- :■•■': '7
C. T
Abstract Fredholm operators are defined and their basic proper- ties are proved. The essential spectruiii of an arbitrary closed operator is considered and its invariance under several kinds of perturbation is proved.
1. Introduction
We first give a complete treat;;ient of Fredholm operators. Most of the results are found in the literature (e.g., [4, 8]), but we have been able to substantially simplify proofs in sev- eral instances. The deepest theorem we raake use of is the closed graph theorem. We are able to avoid completely the concept of the opening betvreen two subspaces.
Next we give a proof of a classical theorem of Weyl [ 13] on the invariance of the essential spectruin under compact perturbations and give two definitions for general closed operators in a Banach space, one of them due to Wolf [ 1^] . We then consider more general types of perturbations. Further results m.ay be found in [ 10] .
Remarks on specific results and }.)ethods are given at the end .
2. FredhoLm Operators
A linear operator A from a Banach space X to a Bane^ch space Y v;ill be called a Fredholm operator if
1. A is closed
2. the domain D(A) of A is dense in X
3. a(A), the dii^iension of the null space N(A) of A, is finite
4. R(A), the range of A, is closed in Y
5. ^(A), the codinension of R(A) in Y, is finite
The index of a Fredholm operator A is defined as
i(A) = a(A) -e(A) .
VJe nox^r vaake several observations. The set of Fredholin operators from X to Y will be denoted by Y(X, Y). 01. If we decompose X into
(1) X = N(A) (i) X' ,
where X is a closed subspace of X, then
(2) llxjl < const. llAxIl
holds for all xfeD(A) ox'. This merely expresses the fact that A restricted to D(A) nx' has as inverse defined every- where on R(A), which is a Banach space. Hence this closed inverse is bounded. ^
02. If A is closed, a{A) < « and (2) holds, then R(A) is closed. For if A:<: — > y, x^'^ D(A) nX', then x^ — > x t X' . Thus x<-. D(A) and Ax = y.
05. If A€l'(X, Y), there is a bounded operator A* from Y to D(A) 0 X' such that
(3) A'A =1 on D(A) nx'
(4) AA' = I on R(A) ,
while A' vanishes on any prescribed comple.aent of R(A) in Y. (Here I denotes the identity operator.) Proof. Obvious.
04. The operator A' also satisfies
(5) A'a = I +F^ on D(A)
( 6 ) AA ' = I + F^ on Y
where the operators F. are bounded and have finite dimensional ranges. '
In fact A A -I vanishes on D(A) .'^x' and hence maps D(A) into the image of N(A). A siiailar statement holds for AA' -I.
05. If A is closed, D(A) dense and there are bounded operators A. from Y to X and compact operators K, on X, Kp on Y such that
(7) A^A = I + K on D(A)
(0) AAg = I + Kp on Y ,
then AcT{X,Y).
Proof. Q (F. Riesz). We must show that R(A) is closed. Suppose
Proof. a(A) <_ a( I + K^) < 00 and e(A) <_ p(I + K^) < «
^0
[x^>'CD(A) oX', 11x^^11 = 1 and Ax^ — > 0. Since A^ is bounded, (I + K, )x — ^ 0. Since (x > is bounded, there is a subse- quence (also denoted by ^x > ) for which K^x. converges to sor.ie element z. Thus x — > -z. Since A is closed zeD(A) and Az = 0, i.e., zfcN(A). Since x' is closed, z '^x' and hence z = 0. But ||zi| = lim ^x^\\ = 1. This shows that (2) holds. Thus R(A) is closed by 02.
We shall also need the following important lemma.
Leimia 1. If At-^(X,Y) and B^ 1'(Z,X), then AB £^r(Z,Y) and
(9) i(AB) = i(A) + i(B) .
7
Proof. Clearly, D(AB) is dense in Z(01). Set
X = R(B) r. K(A) o
R(B) = X ffi' X^
N(A) = X^^ Xp
X = R(B) (? Xg e X^
Then X , Xp, X^ are finite dimensional and X, is closed. Since D(A) is dense in X, vie can take X^ to be contained in D(A). Set dj_ = dij-a X^. Then
a{AB) = a(B) + d P(AB) = P(A) + d^
d^ + dp = a(A) 6,
dg + cl^ = P'ip
These shovj that a(AB) and p(AB) are finite and (9) holds. We must also show that R(AB) is closed. Now R{AB) is the range of A on D(A) 0 X^ . If Ax„ — ^ f for < x„; c D{A) iX^, we have by (2)
iix - X !| < const. 11a.(x -X ) '' m n'' — "^ m n
and since X, is closed, x — * x € X, . Since A is closed, 1 n X
xCD(A) and Ax = f. Thus R(AB) is closed.
It remains to show that AB is closed. Suppose ^z„^c.D(AE), z — ^ z, AEz^ — -^ y. Write
Ez = x(°) +x^'^ , n n n ^
where x^°^c K(A) and x^''^ >-■ x'. Then by (2) xj^"^ — ^ x^"^^^- X',
If llx 11 ^ const., there is a subsequence (also denoted by ^x ^) for v/hich x^°^ — > x^ °^ e ri( a) . Thus Bz — > x^ °^ + x^ ■'■^ and since B is closed zc^D(E) and Ez = x + x . Since A
6
is closed x^°^ + x^-'^tD(A) and A(x^°^ + x^-'"^) = y. Thus z nDiAB) and ABz = y. We nov; show that [x^ ° "| Is bounded.
For if Cn = \kl°^\\ --^ », set u^ = ^~-^x^°^ Then !|u^|| = 1
and hence there is a subseouence (also denoted by ^u„0
<. n -I
such that u — > u r. N(A). Moreover
/ -1 \ -"•
Hence B^ f z ) — > u. Sines Cv^'z — > 0 and B is closed, u = 0. This is impossible since |jujl = lim Hu j] = 1. This completes the proof.
Theorem 1. If A '-^ ^(X, Y) and K is a compact operator from X to Y then (A+K) €?(X,Y) and i{A+K) = i(A).
Proof, By 0^1- there is a bounded operator A' such that (5) and (5) hold. Now
a' (A + K) = I+P-j_+A'K on D(A)
(10) (A+K)A' = I +F2 +KA' on Y .
Since A' is bounded, the operators A K and KA' are compact. We now apply 05 to obtain that (A + X) <J- Y(X, Y) .
Nov.' a' is a bounded operator from Y to D(A). If we equip D(A) vrith the graph norm of A, it becomes a Banach space X and A becomes a Fredholm operator from Y to X. Also AC^^(X,Y) with a(A) and p(A) the sahie. Hence by (j)
(11) i(A) + i(A') = i(I+F^) = 0 (Riesz)
But by ( 10)
i(A+K) + i(A') = id-rFg +1^') = 0 .
Hence i{A+K) = i(A) and the proof is complete.
Theorem 2. For A^'^J{X,Y) there Is an e > 0 such that for any bounded operator T from X to Y with IIt|| < e one has (A + T)P. U'(X,Y), i(A+T) = i(A) and a(A+T) < a(A).
I
Proof. Bj'- Ok there is a bounded operator A satisfying (5) and (6). Then
a'(A + T) = I +F^ +A'T (12) (A+T)A' = I+Fg+TA' .
Take e = I1a' 11""^. Then |Ia't1| < 1 and ITA'II < 1. Thus the operatorsi +A'T a.nd I+TA' are invertible and
(1+ A't)""A'(A + T) = I +(I + a't)"-^Fj_ (A-I-T)A'(I +TA' )"" = I +Fp(I +TA' )""'"
Thus by 05^ (A +T) <i T. By (12) and Theorem 1
i(A+T)+i(A') = id+Fg+TA') = i(I+TA') = 0
, 0 - {
8
Combining this vilth (11) we see that i(A+T) = i(A). It reriiains to prove c(A + T) <_ a(A). By 03, A'(A + T) = I + A'T on D(A) oX' and hence is one-to-one on this manifold. More- over N(A+T) nx = -^O}. For if y is in this set, it is in D(A)oX' and (A+T)y = 0. Thus (I+A'T)y = 0 showing that y = 0. Since
N{A+T) (J X'^X = N(A) (^ X' ,
V7e see that dim N(A +T) <_ dim K(A) and the proof is complete.
In the sequel we shall make continual use of the follov;- ing lemma.
Lemma 2. Let !>. be a Banach space dense in X and let A be a linear operator from X to Y with D(A) = X. If A-^(X,Y), then A€ Y(X,Y).
Proof. Let E be the linear operator from X to X with D(E) = D(A) and Ex = x for xeD(E). One checks easily that E £^(X,X). Thus AE t^?(X,Y) by Lemma 1. But A = AE and the conclusion follows.
An operator B from X to Y is called A-compact if D(B)'2D(A) and for every sequence (x^ !£iD(A) such that
11x^11 + liAx^ll < const.
the seouence --Bx ^) has a convergent subsequence.
Theorem 5. If Ae5;f(x,Y) and B is A-compact, then (A+B)€T(X,Y) and i(A +B) = i(A).
(A)
■-r.'^ X nr
nt ' ' ' Y o&
Proof. If we equip D(A) vjith the graph norm ||x|! + 1|Ax||, it tiecoines a Ba.nach space X (since A is closed). If we now consider A as an operator from X to Y, vje have A£^(X, Y). Moreover B is compact from X to Y. Hence by Theorem 1, (A + B) e Y(X, Y) with i(A + B) = i(A). We now apply Lenmia 2 to obtain (A+B) c^^(x,Y).
Theorem 4. For each A£?(X, Y) there is an e > 0 such that
llBxll < edlxll + llAxIl) , x..D(A) ,
holding for any operator E from X t£ Y with D(B)^D(A) ir.iplies that ( A + B) <£ ^(X, Y), i(A-!-B) = i(A), and g(A + B) ^ a(A).
Proof. Similar to that of Theorem J.
For an arbitrary operator A acting in a Banach space X the Fredholm set of A, denoted by ^», is the set of those complex X for which A- A<^?(X, X). We have by Theorem 1 and 2
Theorem 5. The set ^^ is open and i(A- A) is constant on each component.
Theorem 6. _If K is a compact operator in X, then
^i^+K ^ ^A ^^ i(A+K-A) = i(A-A) there. In addition one also has
Theorem 7. On each component of $., the quantities a( A - A) and P(A - A) are constant except possibly on a discrete set of points v/here they take on larger values.
won t-
0 < s
(A'
10
The proof of Theoren 7 can be made to rest upon Leiima 3. If A and B are linear operators from X to Y
with A€T(X, Y) and B bounded, then there is an s > 0 such tha,t
a{A-AB) is constant for 0 < j>>| < e.
Proof, Assume first that i(A) = 0. Let x,, ...,x, be
a basis for N(A) and y^,...,y, be a basis for some direct
coiLj.)lement of R(A) in Y. Let x',...,x' be a system of bounded
linear functionals on X such that
x!(x^.) = 5.^ (= Kronecker delta) .
One checks easily that the operator defined by
Ax = A:c + > X .(x)y ,
is continuously invertible. Now x is a solution of (A- AB)x = 0 if and only if
(A- AB)x = ^ x:(x)y. .
1=1 ^ ^
For some £, > 0, A - 7\B is continuously invertible for |7\| < e, and hence
= ^~ YZ ^^'(BA-i)"^ YZ ^^•(^)y-i
.1=0 i=l ^ ^
T" x:(x) • T~ ^^(A"^)^x,
i^ ^ 1^ ^
7~~ x:(x)f.(A) ,
i=T ^ ^
11
where f^(A) are knoim vector valued functions in X analytic on |Ai "^ £-!• Thus
<(^) = ir-x'(x)x,'(f,(A))
m ^^ 1 .n 1
or
k
"y [5 . -x'(f.(A))]x!(x) = 0
'-. — T- mi m 1 1
i=l
Conversely, if I-,, •••,Ci, are solutions of
(«) ^t6,.i->^;(fi(^))]5i = o.
then by working back we see that
^ = iz:n^i(^)
i=l ^ ^
is a solution of (A-AB)x = 0. Thus a(A-AB) coincides with the nuraber of linearly independent solutions of (13)« If every coefficient in (13) vanishes identically in JA] < e-|^, there are exactly k linearly independent solutions and a(A-AB) = k for |a| < £-,. Otherwise there is a minor of largest order in the determinant of (13) which does not vanish identically. Since this minor is analytic in A, it can vanish at most at isolated points. Thus there is an e > 0 such that this minor is different from 0 in 0 < |a| < e. The number of independent solutions of (13) is constant on this set and hence
12
the same is true for a(A-AB). Thus the leiroria is proved for i(A) = 0.
Next consider the case i(A) > 0. Let Z be a normed i i(A) I -diraensional space and let Y-, = Y + Z vjith norm |y + z| = |y|f^|z| when y € Y, z tZ. Consider A and B operators from X to Y-, . For these spaces i(A) = 0 and the previous case applies. But a(A- AB) is not changed when we replace Y hy Y, . Thus the lernrna is proved in this case.
If i(A) < 0, let Z be an i( A) -dimensional space and set X, = X(5 Z. Let A and B be extensions of A and B to X, vrhich vanish on Z. Then i(A) = 0 and the first case applies. We then merely observe that a(A - 7\E) = a(A-AB) + i{A) and the proof is complete.
Proof of Theorem 7« Let O be a component of 1.. Let A be any point in O vihere a(A-A) has its minimum value in Q. Then by Le-maa 3 there is a neighborhood about A where a(A- A) has this constant value. Let A-, be any other point in O. It suffices to prove that there is a deleted neighborhood about A-. in v;hich a(A-A) has this minimum value. Connect A, to A by a smooth curve in O. At each point of this curve there is a deleted neighborhood in which a(A- A) is constant (Lerimia J). Since the curve is compact, there is a finite set of such neighborhoods covering it. Since in each deleted neighborhood a(A-A) is constant and each one overlaps with at least one other, they all have the same constant value for a(A- A), namely the minimura value. Q.E.D.
13
In later applications we shall need
Leirjna 4 . If B £ ?( Z, X ) , A is an operator from X to Y with D(A) dense in X, and AB€ Y(Z,Y), then Ae ^(X,Y) .
Proof. Let B be the operator corresponding to B described in Ojj . One easily checks that B ^-^{XjZ) where Z is D(B) under the graph nonn of B. By hjrpothesis AB ^' 'i{Z,Y) and hence ABB' 6: 'i'(x, Y) . But ABB* = A+AF, vrhere F is bounded and of finite rank ( cy-!- ) . Hence AF is a compact operator. We now apply Theorem 1 to conclude that Aey(X, Y),
Lem.-.ia 5. If A'^'T(X, Y), B is an operator from Z to X v/ith D(B) dense in Z, and AB s Y(Z,Y), then B^ ^(Z,X).
Proof. Pollov7ing the same procedure as in the proof of Lemi-a 4, vie have A'AB ^ ^^(Z, X), and hence in ^i'(Z,X). Since A'AB = 3+FB, vjhere F is of finite rank, the operator FB is compact from. Z to X. Hence B •''!'( Z,X) (Theorem 1) and we can apply Le;.ima 2 to conclude that B'5T(Z,X).
3. Weyl s Theorei.i and Generalizations
Let T be a self-ad^'oint operator in a Hilbert space H and let K be a syimvietric, completely continuous operator in H. Then T+K is also self -adjoint . One might ask how ■t(T) and ^(T+K) compare. The ansv.'er v;as given by H. Weyl [ 13] •
Theorem 0. If A is in -^(T) but not in c^(T+K), it m-ust be an isolated eigenvalue of finite '.lultiplicity.
Proof. We note first that since T is self-adjoint, it
1^
is closed, and hence piT)'^^r^. Thus if Ae.fT(T) but not in <rr(T+K), it must he in ^rn = ^^+x' ^°^ otherxrise it could not be in p(T+K). Thus T - p. is a FredhoLn operator for m- in a neighborhood of 7\. Thus ct(T-|j.) is constant in sorr.e deleted neighborhood of A ( LeiiUiia 3)« Since all non-real ix are in p(T), v;e have a(T-ix) = 0 in this deleted neighborhood.
Theorem 9. If a'H-^T) is an isolated ei.genvalue of finite multiplicity;, there is a sy-xaetric, conpletely contin- uous operator K in H such that A t" p ( T + K ) •
Proof. We first shovj tha-t ASi^nr Since. P(T- A) = ci(T- A) < CO, the only thing v;hich must be verified is that R(T-A) is closed in H, i.e., that the inequality
(1^) llxll < C||(T-A)x||
holds for x€D{T) .oN{T-A) (01). Now A is the only point of 7(T) in so-.,:e interval [a,b], where a < A < b. If ^E, ^j is the spectral family for T, then the projection K - E^ maps into N(T- A). But
i(T-A)xI|2 =y^(-^..A)2d|lE^::r
-00
CX5
.:L
> (b-A)2 TclilE xli^ + (a- A)2 fc\\\E:cf
b -03
= (b-A)2|l(l-E^)x|l2 + (a- A)^||E^x|l^ .
15
If X € D(T) n ri(T - A) , ( E, - E )x = 0 and hence ( l4 ) holds, Nov: let h-,^«..,h,. be a basis for N{T- A) satisfying
(h.,hj = a. .
and set
lOc = ^^ {x,h,)h
Then K is bounded, symmetric and of finite ranlt. Thus
T + K- A is a Fredholm operator. Since cx(T+K-A) = p(T+K-7v) = 0,
Aep(T-hK).
We see from Weyl's theorera that the points of o(T) \vhich remain invariant under any syi.metric, completely continuous perturbations are precisely those points which are not isolated eigenvalues of finite multiplicity. The set of such points is called the essential spectrum of T and denoted by <^q(T). Thus Theorems o and 9 can be written as
Theorem 10. If T is self -ad joint and K is syi.raetrlc and completely continuous, then
cr(T + K) = cr(T) .
From the proofs of Theorems 8 and 9 v/e also have Corollary 1. iT (T) is the complement C^^ of $^ in the
coJ:iplex plane.
For ;,iany applications it is Important to generalize to
arbitrary closed operators in Banach space and arbitrary
r:o.':,'r./ (T)
e\ ^m:
■'9a i:.
.-)'-•
l6
compact perturbations. It is no longer true that the invariant points of the spectrum are those which are not isolated eigen- values of finite multiplicity. Wolf [ l4] defines the essential spectru^n by means of property expressed in Corollary 1. We denote the essential spectrum according to this definition by
•y (A).
evr
Definition. For any closed operator A in a Eanach space
We have immediately by Theorem 6
Theorem 11. For any completely continuous operator K,
^ew(A-^K) = cr^^{A) .
We shall also eniploy another definition of essential spectr\i..u
Definition. For a closed operator A ir a Banach space X, '^g--^(A) is the largest subset of cTiA) which remains invariant under arbitrary compact perturbations.
Theorem 12. cr (A) consists of all co'iplex A except
em * ' —
those Afc ^. with i(A- A) = 0.
Proof. If ^^^A ^^^- KA-A) = 0^ then the example given in the proof of Lemma 5 shows that there is an operator F of finite rank for v;hich Af:p(A + F). If i(A - X) ^ 0, then for every compact operator K, i(A+K- A) =1= 0 (Theorem 6) and hence A£:5{A + K). If A^$», then ^^^^a+k ^^^ again A£^(A+K).
IT
We shall nov; give soiae theorer.is on the invariance of
cr (A) and ^ (A) under different types of perturbations, ew eiii
Throughout vje shall assuj.ie that A is a closed operator vfith dense doinain in a Banach space X and B an operator in X with D(B)SD(A).
Theorem 13 . If B is A-conipact, then
(15) ^W^ + S^ = %J^^
ana
(16) - '^pJA+B) = 3- (A)
C.'-U C-.i
Proof. We prove the theorem by showing that ^^ = $_^^g and that i{A-A) = i{A+E-7x) for Afe ^^. If Ae$^, then Ae6.,^ and the index relationship holds (Theore/a 3). Hence ^.C^„ p. If ^^^A+Rj it follows, in particular, that A + B is closed. Since A is closed, we have
(17) ilAxll 1C(11::11 + 11(A+B)::|1) ,
v;hich expresses the fact that A is a closed operator defined everyvjhere on the Banach space D(A-}-B) = D(A) under the graph norm ||x|| + ||(A+B)x!|. Hence it is bounded on this set. Frou (17) it follows that B is ( A+ B)-co'.ipact. We now apply Theorem 3 to A+B with perturbation -B. Thus Afe$^. Hence |^_|_g |^, and the theorem is proved.
18
Theorei-a 1^. If Aep(A) n$^^-g and either (A- A)"-B _or B(A - A)"~ i_s A-conipact then ( 13 ) holcls. If, In addition, l(A-;-B-A) = 0, then (l6) holds.
Proof. We employ the identities
(18) (A + B-M.) - (A- M-)(A- A)"^(A+B- A) = ( n - A) ( A - A) ""^B
(19) (A+B-ix) - (A+E - A)(A- |a)(A- A)"^ = { 1^ - A)B( A - A) "■'-.
Assiime that (A-A)~~B is A-coi!ipa,ct, We equip D(A) v;ith the norm |ix|| + |lA:<:|| + ||Bxil. Under it, D(A) becoines a Banach space X. For if x^^ — * X, Ax^ — ^ y, Bx^ — » z, then x£D(A) and Ax = y since A is closed, while {A+B)x = y+z hecause A+B
is closed. Therefore Bx = z. Now if I-^- ? ly^+g^ A + B - |j. is a
~ \ -1
Fredliolm operator f roi.i X to X. Moreover, (A - A) B is a
compact operator froa X to X, Thus
-."Q
(20) (A- M-)(A- A)~-(A-1-B- A)
•V
is a Fredholm operator from X to X (Theoreiii 1). Nov; (A- A)"^(A+B- A)t ^i^(X,X) (LeM^■;:a 1). Thus {A- •^) e'Y{X,X) ( Lei.iiaa 4 ) and henc e ( A - tx ) £ 1' ( X, X ) , i.e., ii e $^. Therefor ^A+B"^A* Conversely, if I-l € $^, then the operator (20) is in 'I'(X,X). Nence the sa.iie is true of A+B-[x, i.e., lael^^g. Thus 5y^ = ^A+B* '^^'^^ proves (15). If KA+B-A) = 0, v;e have by (l8) 1/ A ^^ ' ' ^ ''■'^ ' '''^
i(A+E-|x) = i(A- ix) ■ , c^>)-_ y'(- )'-^'^'V)
19
and hence (l6) holds.
Theorem 15 . !£. B i£ A -compact and ^ '^^ a ''^ ^A+B' '^!"^^^
(15) holds. If, In addition, i(A-A) = i(A + B-A), then
(16) holds.
Proof, Consider the norm
llUil = 11x11 + llAxll + iBxll + llA^xll + llBAxll
p 2 I''
on D(A~). We claim that D(A ) equip.ed with this norm
becomes a, Banach space X. In fact if x — =^ x. Ax — ^^^ y,
Bx — ^ z, A X — » u, iiAx — * V, then XfeD(A) since A is closed n n n
and A:z = y, v/hile y feD(A) and Ay = u for the same reason. Thus xfcDCA"). Since A+B is closed, (A+ B)x = y+ z and hence Bz = z. The same reasoning gives (A+ B)Ax = u + v, whence BAx = V. Now suppose M- ^ |i„ . Then the operator (A + E - A)(A -M.) ^^(X,X) by Lemma 1 and hence is in ^r(X,X). Hov/ever, when we consider B as an operator from X to X, it becomes a compact operator. Since
(21) (A+B-A)(A-ia) - ( A +B - 11) ( A- A) = (A-z)B ,
vre see that ( A -^ B - |i) (A - A) is a Fredholm operator from X to X (Theorem 1). Thus it is in ^'(X, X) (Lemia 2). Since (A- A) ^'l'(X, X) v;e can apply Lemjiia k to obtain that (A + B - M-) '^'•^(X, X). Hence 1^— I^s^.l^* The same reasoning applied in the opposite direction gives $^ = ^A-I-B' Moreover for M- 1 § A we ha,ve by (21), Lerana 1 s-nd Theorem 1
xAi!
'iOn Qldi 1
■'■. A
V -*-;■-
.■q;j a
i-A)
■,'-: t
20
i(A + B-A) + i(A-!x) = i(A+B-|x) + i(A - A) Hence
i(A - |x) = i(A+ B- M-) .
Thus i^'^(A + B) = cr^ (A) and the proof is coimlete.
We see from the proof of Theorem 15 that the assumption
2 that B is A -compact can be considerably weakened.
4 . Remarks
Rl. In the Russian literature operators satisfying properties 1, 5 -5 are called ^-operators, with the \ standing for Fredholm (cf. ['']). The term Fredholra operator seems to be reserved for ^-operators having index 0. We have added assui?iption 2 fcr convenience and do not differentiate on the basis of index.
R2. Observations 04 and 05 are due to Atkinson [ l] for bounded operators (cf, also [11,12]).
RJ. Leroma 1 is also dut to Atkinson [ l] for bounded operators. For unbounded operators it is due to Gohberg [3]) although there seems to be a gap in his proof (cf. A.I"i.S. translation of [4]). The first part of our proof follows that of [4], but our proof that R(AB) is closed is different. Our proof that AB is a closed operator was taken from Kato [8]. Other proofs may be found in [6,8].
21
R4 . For the histories of Theorems 1 and 2 see [4], Our proofs "borrow ideas from Seeley [ 12] . Our proof of a(A+T) < a(A) appears to be much simpler than any fouv^d in the literature.
•R5. The idea for Lemma 2 and its proof came from Kato [8]
r6. Theorems 3 and 4 as well as the device employed in obtaining them from Theorems 1 and 2 are due to Nagy [ 9] •
R7« Generalizations of Theorems 5-7 can be found in [8,7]. Cur proof of Theorem 7 is taken directly from [4],
R8. Leranas 4 and 5 are apparently new.
R9. The term essential spectrum originated in [53 vrtiere it was applied to self-adjoint problems for ordinary differ- ential equations on a half-line. In this case the essential spectrum is that part of the spectrum which remains invariant under changes in the boundary conditions. Browder [2] has proposed still another definition.
RIO. Theorems 12, Ik, 15 are apparently new. Similar results may be found in [2, l4].
Bibliography
1. F. V, Atkinson, Mat. Sb. 28(70) { 1951)5 - 1^ •
2. F. E. Browder, Math, Ann. l42( 1961)22 - IJO.
5. I. C. Gohberg, Mat. Sb. 53(75 )( 1953 )195 - 198.
^. I. C. Gohberg and M, G. Krein, Usp. Mat. Nauk 12(1957) 43-118; Transl. A.M.S., 13( 19^0) 185 - 2o4 .
5. Philip Hartman and A.urel Wintner, Aiiier, J. Math. 72 (1950)543-552.
6. J. T. Joichi, dissertation, Illinois, 1959«
7. Shmuel Kaniel and Martin Schechter, Coimn. Pure Appl. Math., 16(1965)423-448.
8. Tosio Kato, J. Analyse Math., 6(1958)261-522.
9. Bela Szo-Nagy, Acta Sci. Math. Szeged 14(1951)125-137.
10. Martin Schechter, Invariance of the essential spectrum, to appear.
11. J. T. Schwartz, Cor;m. Pure Appl. Math., 15(1962)75-90.
12. R. T. Seeley, J. Math. Ann. Appl., 7(1965)289-509.
15. Herman Weyl, Rend. Circ. Mat. Palermo, 27(1909)573-392.
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BORROWER S NAME
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